Finite Group Databasebeta

← #28610718

Subgroups — #28610718

Order 15104.

850 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837680C211yesmaximal, characteristic
2#21427183C211yesmaximal, characteristic, complemented
3#21427001C211yesmaximal, characteristic
4#21425067C211yesmaximal, characteristic, complemented
5#21424863C211yesmaximal, characteristic, complemented
6#21424834C211yesmaximal, characteristic, complemented
7#21424828C211yesmaximal, characteristic, complemented
8#21424687C211yesmaximal, characteristic, complemented
9#21424667C211yesmaximal, characteristic, complemented
10#21424359C211yesmaximal, characteristic
11#21424343C211yesmaximal, characteristic
12#21423972C211yesmaximal, characteristic, complemented
13#21423923C211yesmaximal, characteristic
14#102×Dic59C211yesmaximal, characteristic
15C59×#1762C211yesmaximal, characteristic
16C2×#14273726C2211yescharacteristic
17C2×D4×Dic59C2211yescharacteristic
18C2×D4×Dic59C2211yescharacteristic
19#16837680C418yescomplemented
20C2×#14273661C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×Dic59⋊C8C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273637C2211yescharacteristic
28C2×C8×Dic59C2211yescharacteristic
29#16837308—21no—
30#16837306—21no—
31#16837301C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837299—21no—
35#16837298C2211yescharacteristic
36#16837275—21no—
37#16837274—21no—
38#16837267C2211yescharacteristic
39#16837260—21no—
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4×Dic59⋊C4—21no—
43C4×Dic59⋊C4C2211yescharacteristic
44C2×C59⋊#95—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837157C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×C4×D4×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C2×#97×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×Dic118—21no—
61C23×Dic59—21no—
62C23×Dic59—21no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273726—22no—
66#14273726C2×C414yescomplemented
67D4×Dic59—22no—
68D4×Dic59—22no—
69D4×Dic59—41no—
70D4×Dic59C2×C414yes—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4D411yescharacteristic
81C2×Dic59⋊C4D411yescharacteristic
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59C2311yescharacteristic
87C2×C4×Dic59C2311yescharacteristic
88#14273661C2×C414yescomplemented
89#14273660—41no—
90#14273660C2×C414yescomplemented
91#14273659—41no—
92Dic59⋊C8C2×C414yescomplemented
93#14273648—22no—
94#14273648C2×C414yes—
95#14273637—22no—
96C8×Dic59—41no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95D411yescharacteristic
106C59⋊#95D411yescharacteristic
107C22×D4×C59C2311yescharacteristic
108C23×C4×C59—21no—
109C22×C8×C59C2311yescharacteristic
110C22×C8×C59C2311yescharacteristic
111C4×D4×C59—22no—
112C4×D4×C59—41no—
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C22⋊C4×C59—21no—
117C2×C42×C59—21no—
118C2×C42×C59C2311yescharacteristic
119#102×C59C2×C414yes—
120#102×C59Q814yescomplemented
121#97×C59—41no—
122#95×C59—21no—
123C24×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59C22×C412yes—
128C2×D4×C59C2×Q812yes—
129C2×D4×C59#4112yes—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—42no—
134C22×C4×C59—42no—
135C22×C4×C59—81no—
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2411yescharacteristic
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—24no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—42no—
145C22×Dic59—42no—
146C22×Dic59—81no—
147C22×Dic59—81no—
148C22×Dic59—81no—
149C22×Dic59C2×D411yescharacteristic
150C22×Dic59C2×D411yescharacteristic
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153C2×Dic118—24no—
154C2×Dic118—81no—
155C2×Dic118—81no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59C22×C412yes—
161C2×C8×C59C2×Q812yes—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59C22×C412yes—
167C4⋊C4×C59C2×Q812yes—
168C22⋊C4×C59—42no—
169C42×C59—21no—
170C42×C59—21no—
171C42×C59—81no—
172C22⋊Dic59—24no—
173C22⋊Dic59—42no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—81no—
181C4⋊Dic59C22×C412yes—
182C4⋊Dic59C22×C412yes—
183Dic59⋊C4—24no—
184Dic59⋊C4—22no—
185Dic59⋊C4—41no—
186Dic59⋊C4—81no—
187Dic59⋊C4—81no—
188Dic59⋊C4—81no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59—42no—
192C4×Dic59—81no—
193C4×Dic59C22×C412yes—
194C4×Dic59#4112yes—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8C22×C412yes—
201C23×C59—22no—
202C23×C59—22no—
203C23×C59—22no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59C22×D411yescharacteristic
209D4×C59—22no—
210D4×C59—22no—
211D4×C59—22no—
212D4×C59C4×Q814yes—
213C2×C4×C59—22no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—22no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—84no—
229C2×C4×C59C23×C411yescharacteristic
230C2×C4×C59C22×Q811yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59H(5,𝔽2)11yescharacteristic
234C2×C4×C592−1+411yescharacteristic
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—22no—
242C2×Dic59—22no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—84no—
257C2×Dic59—84no—
258C2×Dic59C4×D412yes—
259C2×Dic59C4×D412yes—
260C2×Dic59#13712yes—
261Dic118—84no—
262C8×C59—22no—
263C8×C59—22no—
264C59⋊C8—22no—
265C59⋊C8—41no—
266#22979—591nomaximal, Sylow
267C22×C59—24no—
268C22×C59—42no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—42no—
274C22×C59C2×C4×D411yescharacteristic
275C22×C59#54611yescharacteristic
276C22×C59D4×Q811yescharacteristic
277C22×C59C2×#13711yescharacteristic
278C22×C59#57511yescharacteristic
279C22×C59#57611yescharacteristic
280C22×C59#57611yescharacteristic
281C4×C59—42no—
282C4×C59—42no—
283C4×C59—41no—
284C4×C59—41no—
285C4×C59—81no—
286C4×C59C2×C4×Q811yescharacteristic
287C4×C59#52011yescharacteristic
288C4×C59#55211yescharacteristic
289C4×C59#55411yescharacteristic
290Dic59—22no—
291Dic59—22no—
292Dic59—22no—
293Dic59—42no—
294Dic59—42no—
295Dic59—41no—
296Dic59—81no—
297Dic59—81no—
298C2×#474—591no—
299#2369—591no—
300#2349—591no—
301#1993—591no—
302#1949—591no—
303#1943—591no—
304#1942—591no—
305#1921—591no—
306#1918—591no—
307#1864—591no—
308#1862—591no—
309#1771—591no—
310#1762—591no—
311#1762D5911yescharacteristic
312C4×#102—591no—
313C2×C59—44no—
314C2×C59#236011yescharacteristic
315C2×C59C4×#13711yescharacteristic
316C2×C59#294811yescharacteristic
317C2×C59#331411yescharacteristic
318C2×C59#331911yescharacteristic
319C2×C59#338511yescharacteristic
320C2×C59#338911yescharacteristic
321C2×C4⋊Q8—591no—
322C2×C4×D4—591no—
323C2×C4×D4—591no—
324C2×C4×D4C2×D5911yescharacteristic
325#474—598no—
326C2×#107—591no—
327C2×#107—591no—
328C2×#107—591no—
329C2×#106—591no—
330C2×C4⋊C8—591no—
331C2×#102—591no—
332C2×#102—591no—
333C2×#102C2×D5911yescharacteristic
334C2×#102C2×D5911yescharacteristic
335C2×#97—591no—
336C2×#97C2×D5911yescharacteristic
337C2×C4×C8—591no—
338#391—1181no—
339#391—1181no—
340#389—591no—
341#389—591no—
342#389—591no—
343#389—591no—
344#389—1181no—
345#385—1181no—
346#385—1181no—
347#382—591no—
348#380—1181no—
349C4×C4⋊C4—591no—
350C4×C4⋊C4—591no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—1181no—
353C4×C4⋊C4C2×D5911yescharacteristic
354C2×#95—1181no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340C2×D5911yescharacteristic
362#340C2×D5911yescharacteristic
363#336—591no—
364#336—591no—
365C59#2297911yescharacteristic, complemented, Sylow
366C22×Q8—1181no—
367C22×D4C22×D5911yescharacteristic
368C23×C4—21no—
369C23×C4—1181no—
370C23×C4—1181no—
371C22×C8—591no—
372C22×C8—591no—
373C22×C8C22×D5911yescharacteristic
374C22×C8C22×D5911yescharacteristic
375C4⋊Q8—594no—
376C4⋊Q8—1182no—
377C4×D4—22no—
378C4×D4—41no—
379C4×D4—594no—
380C4×D4—1182no—
381C4×D4—1182no—
382C4×D4—2361no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4C22×D5911yescharacteristic
395C2×C4⋊C4C22×D5911yescharacteristic
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1181no—
399C2×C22⋊C4—1181no—
400C2×C42—21no—
401C2×C42—591no—
402C2×C42—591no—
403C2×C42—1181no—
404C2×C42—1181no—
405C2×C42C22×D5911yescharacteristic
406#107—594no—
407#107—594no—
408#107—2361no—
409#106—2361no—
410C4⋊C8—594no—
411#102—594no—
412#102—1182no—
413#102Dic11814yescomplemented
414#102C4×D5914yes—
415#97—41no—
416#97—1182no—
417C4×C8—2361no—
418#95—21no—
419#95—591no—
420#95—591no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429C24—21no—
430C2×Q8—1184no—
431C2×Q8—4721no—
432C2×Q8—4721no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4C2×Dic11812yes—
437C2×D4C2×C4×D5912yes—
438C2×D4#1174568512yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—81no—
445C22×C4—591no—
446C22×C4—591no—
447C22×C4—591no—
448C22×C4—591no—
449C22×C4—1184no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4—1181no—
454C22×C4—4721no—
455C22×C4—4721no—
456C22×C4—4721no—
457C22×C4—2362no—
458C22×C4—2362no—
459C22×C4D4×D5911yescharacteristic
460C22×C4D4×D5911yescharacteristic
461C22×C4C23×D5911yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—592no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8C2×Dic11812yes—
473C2×C8C2×C4×D5912yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—592no—
479C4⋊C4—592no—
480C4⋊C4—1182no—
481C4⋊C4—1184no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—4721no—
489C4⋊C4—4721no—
490C4⋊C4—4721no—
491C4⋊C4—4721no—
492C4⋊C4—2361no—
493C4⋊C4C2×Dic11812yes—
494C4⋊C4C2×C4×D5912yes—
495C22⋊C4—42no—
496C22⋊C4—1184no—
497C22⋊C4—2362no—
498C42—21no—
499C42—21no—
500C42—81no—
501C42—592no—
502C42—592no—
503C42—1181no—
504C42—1181no—
505C42—4721no—
506C42—2362no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C2×D4×D5911yescharacteristic
515Q8—4724no—
516D4—22no—
517D4—22no—
518D4—22no—
519D4C4×Dic11814yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
536C2×C4—592no—
537C2×C4—592no—
538C2×C4—592no—
539C2×C4—1181no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—1181no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1182no—
549C2×C4—1182no—
550C2×C4—4724no—
551C2×C4—4724no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4C22×C4×D5911yescharacteristic
563C2×C4C22×Dic11811yescharacteristic
564C2×C4#41×D5911yescharacteristic
565C2×C4C2×#1174568511yescharacteristic
566C2×C4#1427377211yescharacteristic
567C2×C4#1427377411yescharacteristic
568C8—22no—
569C8—22no—
570C8—1182no—
571C8—2361no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C4×D4×D5911yescharacteristic
580C22#1683804211yescharacteristic
581C22D4×Dic11811yescharacteristic
582C22#137×D5911yescharacteristic
583C22#1683819811yescharacteristic
584C22#1683820611yescharacteristic
585C22#1683820611yescharacteristic
586C4—41no—
587C4—41no—
588C4—42no—
589C4—42no—
590C4—81no—
591C4—1182no—
592C4—1182no—
593C4—1182no—
594C4—4721no—
595C4—4721no—
596C4—2361no—
597C4—2362no—
598C4—2362no—
599C4C2×C4×Dic11811yescharacteristic
600C4#1683788111yescharacteristic
601C4#1683808511yescharacteristic
602C4#1683811111yescharacteristic
603C2—44no—
604C2#2142710211yescharacteristic
605C2#2143291811yescharacteristic
606C2#2143294511yescharacteristic
607C2#2143617811yescharacteristic
608C2#2143625311yescharacteristic
609C2#2143700711yescharacteristic
610C2#2143705511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1762C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1762D59
hypocenterC59#22979
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21427183C2semidirect
4. #21425067C2semidirect
5. #21424863C2semidirect
6. #21424834C2semidirect
7. #21424828C2semidirect
8. #21424687C2semidirect
9. #21424667C2semidirect
12. #21423972C2semidirect
19. #16837680C4semidirect
66. #14273726C2×C4semidirect
88. #14273661C2×C4semidirect
90. #14273660C2×C4semidirect
92. Dic59⋊C8C2×C4semidirect
120. #102×C59Q8semidirect
365. C59#22979semidirect
413. #102Dic118semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168376802
#214271832
#214270012
#214250672
#214248632
#214248342
#214248282
#214246872
#214246672
#214243592
#214243432
#214239722
#214239232
#102×Dic592
C59×#17622
#2297959